Robust and Fast Algorithm for a Circle Set Voronoi Diagram in a Plane

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Robust and Fast Algorithmfor a Circle Set
Voronoi Diagramin a Plane
The 2001 International Conference on
Computational Science San Francisco, CA, USA May
28, 2001 D.-S. Kim, D. Kim, K. Sugihara and
J. Ryu Hanyang University, Seoul, Korea
University of Tokyo, Tokyo, Japan
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Problem Definition

• Given VD(P)
• Find VD(C)
• Get the topology of VD(C) from VD(P)
• Update geometric values

Basic Idea
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Introduction
Point Set Voronoi Diagram VD(P)
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• Point set Voronoi diagram VD(P)
• Well understood
• Efficient/robust algorithm exists
• Excellent code is available
• www.simplex.t.u-tokyo.ac.jp/sugihara

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Circle Set Voronoi Diagram

• Hyperbolic arc
• Star shaped polygon

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Previous Works

• Kirkpatrick (79)
• line seg./polygon, DC
• Lee Drysdale (81)
• line seg./polygons/circles, DC, O(nlog2n)
• Sharir (85)
• circles, DC, O(nlog2n)
• Yap (87)
• line seg./circles, DC, O(nlogn)
• Fortune (87)
• points/line seg./circles, line sweeping, O(nlogn)
• Sugihara (93)
• approximation
• Gavrilova Rokne (99)
• swap condition of dynamic VD(C)

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Flipping Condition

• Two CC exist (No Flip)

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Flipping Condition

• Two CC exist (Flip)

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Flipping Condition

• One CC exists (Flip)

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Flipping Condition

• One CC exists (No Flip)

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Flipping Condition

• No CC exists (No Flip)

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Summary of Flipping Condition
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Two-edge Face
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New Born Edges Vertices
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Removed Edges Vertices
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Treatment

• Create 4 fictitious generators

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Time Complexity O(n2)
Delaunay (P)
Delaunay (C)
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Procedure Composition

• Generator-by-generator approach
• greedy generator
• Edge-by-edge approach
• flippancy edge

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Geometry Update

• Vertex geometry
• Center of a circumcircle of three generators.
• Apollonius Tenth Problem
• Edge geometry
• Bisector between two generators.

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Edge Geometry

• Hyperbola (or line)
• Rational quadratic Bézier curve
• Conditions to determine RQB.
• Two end points given by Voronoi vertices
• Two tangents at the end points
• One passing point

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Tangent Line Passing Point
Tangent Vector Angle bisector
Passing point
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Experiments

• 3,500 random circles

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1,000 random circles
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800 random circles on a spiral one large circle
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400 random circles on a circle one large circle
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400 equi-radii circles on a circle one large
circle.
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400 random circles on a circle
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Number of Flips (1,200 generators)
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Conclusions

• New algorithm for VD(C)
• Simple to code
• As robust as VD(P) algorithm
• Fast
• no removal of data object
• no trimming
• Extendable to other generalized problems